Radiative Transfer Theory at Optical wavelengths applied to vegetation canopies: part 2 UoL MSc Remote Sensing Dr Lewis [email protected].

Radiative Transfer Theory at Optical wavelengths applied to vegetation canopies: part 2

UoL MSc Remote Sensing

Dr Lewis [email protected]

Radiative Transfer equation

• Used extensively for (optical) vegetation since 1960s (Ross, 1981)

• Used for microwave vegetation since 1980s

Radiative Transfer equation

• Consider energy balance across elemental volume

• Generally use scalar form (SRT) in optical• Generally use vector form (VRT) for microwave

z

Pathlength l

z = l cos l

Medium 1: air

Medium 2: canopy in air

Medium 3:soil

Path of radiation

Scalar Radiative Transfer Equation

• 1-D scalar radiative transfer (SRT) equation– for a plane parallel medium (air) embedded with a low

density of small scatterers– change in specific Intensity (Radiance) I(z,) at depth z

in direction wrt z:

( ) ( )zJzIz

zIse ,),(

,Ω+Ω−=

∂

Ω∂κμ

Scalar RT Equation

• Source Function:

• - cosine of the direction vector (with the local normal

– accounts for path length through the canopy

• e - volume extinction coefficient

• P() is the volume scattering phase function

( ) ′′→′= ∫ dIzPzJ S ),();,(,4π

( ) ( )zJzIz

zIse ,),(

,Ω+Ω−=

∂

Ω∂κμ

Extinction Coefficient and Beers Law

• Volume extinction coefficient:– ‘total interaction cross section’– ‘extinction loss’– ‘number of interactions’ per unit length

• a measure of attenuation of radiation in a canopy (or other medium).

e

( ) leeIlI κ−= 0Beer’s Law

Extinction Coefficient and Beers Law

€

I l( ) = I0e−κ e l

I

eIdl

dI

e

le

e

−=

−= −

No source version of SRT eqn( ) ( )zJzI

z

zIse ,),(

,Ω+Ω−=

∂

Ω∂κμ

Optical Extinction Coefficient for Oriented Leaves

• Volume extinction coefficient:

• ul : leaf area density – Area of leaves per unit volume

• Gl : (Ross) projection function€

e Ω( ) = ulGl Ω( )

( ) ( ) lllll dgG ΩΩ⋅ΩΩ=Ω ∫+ππ 22

1

Optical Extinction Coefficient for Oriented Leaves

( ) ( ) lllll dgG ΩΩ⋅ΩΩ=Ω ∫+ππ 22

1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80 90

G_l(theta)

zenith angle / degrees

spherical planophile erectophileplagiophile extremophile

Optical Extinction Coefficient for Oriented Leaves

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80 90

G_l(theta)

zenith angle / degrees

spherical planophile erectophileplagiophile extremophile

• range of G-functions small (0.3-0.8) and smoother than leaf inclination distributions;

• planophile canopies, G-function is high (>0.5) for low zenith and low (<0.5) for high zenith;

• converse true for erectophile canopies;• G-function always close to 0.5 between 50o and 60o

• essentially invariant at 0.5 over different leaf angle distributions at 57.5o.

Optical Extinction Coefficient for Oriented Leaves

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80 90

G_l(theta)

zenith angle / degrees

spherical planophile erectophileplagiophile extremophile

• so, radiation at bottom of canopy for spherical:

• for horizontal:

€

−ulG Ω( )

μdz =

LG Ω( )μz= 0

z=− H

∫

( )

LLG

eIeI5.0

00

−Ω

−

=

LeI −=

A Scalar Radiative Transfer Solution

• Attempt similar first Order Scattering solution– in optical, consider total number of interactions

• with leaves + soil

• Already have extinction coefficient:

( ) ( )= Guleκ

SRT

• Phase function:

• ul - leaf area density;

• ’ - cosine of the incident zenith angle• - area scattering phase function.

( ) ( )→′′

=→′ luPμ

1

SRT

• Area scattering phase function:

• double projection, modulated by spectral terms l : leaf single scattering albedo

– Probability of radiation being scattered rather than absorbed at leaf level– Function of wavelength

€

′→ ( ) =1

4πωlgl Ω l( )Ω ⋅Ω l ′ Ω ⋅Ω l dΩ l

4π

∫

SRT

( )

( ) ( )

( ) ( )

( )( ) ( )

( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+→′

+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

+

0

00

0

00

1

,

,

00

0

000

μμ

μμ

μμ

μμ

μμ

δρ

s

ss

s

ss

GGL

ss

sssoil

GGL

s

eGG

I

Ie

zI

SRT: 1st O mechanisms

• through canopy, reflected from soil & back through canopy

( )

( ) ( )

( ) ( )

( )( ) ( )

( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+→′

+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

+

0

00

0

00

1

,

,

00

0

000

μμ

μμ

μμ

μμ

μμ

δρ

s

ss

s

ss

GGL

ss

sssoil

GGL

s

eGG

I

Ie

zI( ) ( )

( )0,0

00

ΩΩ⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω+Ω−

ssoil

GGL

s

ss

e ρμμ

μμ

1. 2.

SRT: 1st O mechanisms

( )

( ) ( )

( ) ( )

( )( ) ( )

( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+→′

+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

+

0

00

0

00

1

,

,

00

0

000

μμ

μμ

μμ

μμ

μμ

δρ

s

ss

s

ss

GGL

ss

sssoil

GGL

s

eGG

I

Ie

zI

1. 2.

( )( ) ( )

( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+→′ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−

0

00

100

μμ

μμ

μμs

ss GGL

ss

eGG

Canopy only scattering

Direct function of Function of gl, L, and viewing and illumination angles

1st O SRT

• Special case of spherical leaf angle:

( )

( ) ( ) γργγγπτρ

cos3

cossin3

5.0

lll

G

+−+

=Ω→Ω′Γ

=Ω

′⋅=γcos

Multiple Scattering

LAI 1

Scattering order

Contributions to reflectance and transmittance

Multiple Scattering

LAI 5

Scattering order

Contributions to reflectance and transmittance

Multiple Scattering

LAI 8

Scattering order

Contributions to reflectance and transmittance

Multiple Scattering

– range of approximate solutions available– Recent advances using concept of recollision

probability, p• Huang et al. 2007

Q0

s

i0

i0=1-Q0

p

s1=i0 (1 – p)

p: recollision probability: single scattering albedo of leaf

• 2nd Order scattering:

pi

s

i

s

1

2 =

i0

i0 p

2 i0 p(1-p)

( ) ( ) ( ) L+−+−+−= 232

0

111 pppppi

sωωω

( )pi

s−= 1

0

1 ω

( ) ( ) ( ) L+−+−+−= 232

0

111 pppppi

sωωω

( )[ ]L+++−= 22

0

11 pppi

sωωω

( )

p

p

i

s

−

−=

1

1

0

‘single scattering albedo’ of canopy

( )

p

p

i

s

−

−=

1

1

0

( ) ( )λλ

pns −

=1

1 Average number of photon interactions:The degree of multiple scattering

p: recollision probability

( ) ( )( )λλλα

p−

−=

1

1Absorptance

Knyazikhin et al. (1998): p is eigenvalue of RT equationDepends on structure only

• For canopy:

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10

LAI

pcanopy

Smolander & Stenberg RSE 2005

( )( )bcanopy kLAIpp −−= exp1max

pmax=0.88, k=0.7, b=0.75Spherical leaf angle distribution

Canopy with ‘shoots’ as fundamental scattering objects:

canopyi

s⎟⎟⎠

⎞⎜⎜⎝

⎛

( )shootcanopy

shootcanopycanopy p

p

ω

ωω

−

−==

1

1

Clumping

( )shootcanopy

shootcanopycanopy

canopyp

p

i

s

ω

ωω

−

−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

1

1

0

Canopy with ‘shoots’ as fundamental scattering objects:

( )needleshoot

needleshootshoot

shootp

p

i

s

ω

ωω

−

−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

1

1

0

( )needle

needlecanopy

canopyp

p

i

s

ω

ωω

2

2

0 1

1

−

−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

( ) shootcanopycanopy pppp −+= 12

p2

pcanopy

Smolander & Stenberg RSE 2005

• pshoot=0.47 (scots pine)

• p2<pcanopy

• Shoot-scale clumping reduces apparent LAI

Other RT Modifications

• Hot Spot– joint gap probabilty: Q

– For far-field objects, treat incident & exitant gap probabilities independently

– product of two Beer’s Law terms

′′′

→′

)G( + )G( L -

= )Q( e

RT Modifications

• Consider retro-reflection direction:– assuming independent:

– But should be:

)G( L 2

-

= )Q(Ω

Ω→Ω e

)G( L

-

= )Q(Ω

Ω→Ω e

RT Modifications

• Consider retro-reflection direction:– But should be:

– as ‘have already travelled path’– so need to apply corrections for Q in RT

• e.g.

)G( L

-

= )Q(Ω

Ω→Ω e

),C( )P( )P( = )Q( ′′→′

RT Modifications

• As result of finite object size, hot spot has angular width– depends on ‘roughness’

• leaf size / canopy height (Kuusk)• similar for soils

• Also consider shadowing/shadow hiding

Summary

• SRT formulation– extinction– scattering (source function)

• Beer’s Law– exponential attenuation – rate - extinction coefficient

• LAI x G-function for optical

Summary

• SRT 1st O solution– use area scattering phase function– simple solution for spherical leaf angle– 2 scattering mechanisms

• Multiple scattering– Recollison probability

• Modification to SRT:– hot spot at optical